Lecture 17 - Interference
Source: Forbes (2022)
Source: Centola (2010)
Source: Hemming et al (2014)
Assigned treatment
| Market | Week 1 | Week 2 | Week 3 |
|---|---|---|---|
| 1 | 01 | 11 | 11 |
| 2 | 00 | 00 | 01 |
| 3 | 00 | 01 | 11 |
| 4 | 00 | 00 | 01 |
| 5 | 00 | 00 | 00 |
| 6 | 01 | 11 | 11 |
| 7 | 00 | 00 | 00 |
| 8 | 00 | 01 | 11 |
Observed outcomes
| Market | Week 1 | Week 2 | Week 3 |
|---|---|---|---|
| 1 | 7 | 9 | 4 |
| 2 | 7 | 5 | 7 |
| 3 | 1 | 2 | 10 |
| 4 | 4 | 3 | 10 |
| 5 | 3 | 3 | 3 |
| 6 | 10 | 8 | 10 |
| 7 | 2 | 3 | 4 |
| 8 | 3 | 1 | 3 |
Probabilities of assignment to treatment condition
| Treatment Condition | Week 1 | Week 2 | Week 3 |
|---|---|---|---|
| Pr(00) | 0.75 | 0.50 | 0.25 |
| Pr(01) | 0.25 | 0.25 | 0.25 |
| Pr(11) | 0 | 0.25 | 0.50 |
\[ \begin{aligned} \widehat{E}[Y_{01} - Y_{00}] &= \frac{\frac{7 + 10}{0.25} + \frac{2 + 1}{0.25} + \frac{7 + 10}{0.25}}{\frac{2}{0.25} + \frac{2}{0.25} + \frac{2}{0.25}} \\ &- \frac{\frac{7 + 1 + 4 + 4 + 3 + 2 + 3}{0.75} + \frac{5 + 3 + 3 + 3}{0.50} + \frac{3 + 4}{0.25}}{\frac{6}{0.75} + \frac{4}{0.50} + \frac{2}{0.25}} = 2.72. \end{aligned} \]
\[ \begin{aligned} \widehat{E}[Y_{11} - Y_{00}] &= \frac{\frac{9 + 8}{0.25} + \frac{4 + 10 + 10 + 3}{0.50}}{\frac{2}{0.25} + \frac{4}{0.50}} \\ &- \frac{\frac{5 + 3 + 3 + 3}{0.50} + \frac{3 + 4}{0.25}}{\frac{4}{0.50} + \frac{2}{0.25}} = 4.13. \end{aligned} \]